# When Quantum Mechanics is thrashed by non-physicists #1

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Historically, we know that above any other discipline of physics, quantum mechanics has attracted other types of scientists, such as mathematicians, chemists, specialists in information technology, but also, philosophers of science. While most of their historical contributions are really valuable (think von Neumann or Weyl), it is with considerable regret that I sense that nowadays, in the huge sea of 'which interpretation is better' viewpoints, their contributions are rather distructive, (perhaps I exaggerate here) or at least very challenging.
I recently bumped into an article on the preprints server:

http://arxiv.org/abs/1412.2701v1

thrashing the idea that a (unit) vector of a Hilbert space can represent the state of a physical quantum system. One of the authors is a philosopher, the other a mathematician and. surprisingly, throughout the paper the mathematics of finite-dim. vector spaces (known to be improper for QM) is used. I don't think, however, that the lack of rigor in maths can be the point which turns their paper from a correct one into a wrong one.

Have a read of it, please, and tell me where they go wrong, if anywhere. (As a joking sidepoint: Is there a catholic formalism of QM, too, because the literature is flooded with protestant ones challenging the orthodox one??:D)

Thanks,
:)

Demystifier

ShayanJ
Gold Member
, it is with considerable regret that I sense that nowadays, in the huge sea of 'which interpretation is better' viewpoints, their contributions are rather distructive, (perhaps I exaggerate here) or at least very challenging.

I don't think I know enough physics to comment on the technical parts. But I should say I don't accept that such papers are "destructive" or anything that bad! But "challenging", I accept. But I'm surprised that you're using this word as it is a weaker form of destructive! Of course if physicists overcome such challenges, it doesn't mean a waste of time or effort because they surely gained a deeper understanding. On the other hand, if physicists can't come up with an answer, then it means the those challenges should themselves point to a deeper thing.
But I suspect you mean the authors have a misunderstanding about QM. Which I don't think is probable!

Here I like to quote some part of Feynman's Nobel lecture:
Richard Feynman said:
If every individual student follows the same current fashion in expressing and thinking about electrodynamics or field theory, then the variety of hypotheses being generated to understand strong interactions, say, is limited. Perhaps rightly so, for possibly the chance is high that the truth lies in the fashionable direction. But, on the off-chance that it is in another direction - a direction obvious from an unfashionable view of field theory - who will find it? Only someone who has sacrificed himself by teaching himself quantum electrodynamics from a peculiar and unusual point of view; one that he may have to invent for himself. I say sacrificed himself because he most likely will get nothing from it, because the truth may lie in another direction, perhaps even the fashionable one.

mheslep
bhobba
Mentor

Am I imagining something or is the opening line totally 'whaco':
'In this paper we derive a theorem which proves that the physical interpretation implied by the first postulate of quantum mechanics (QM) is inconsistent with the orthodox formalism.'

How can a postulate be inconsistent with the formalism it is part of?

I am not too worried about the finite dimensional thing because I view QM in the Rigged Hilbert Space formalism and think of finite dimensional states as the physical ones while the usual ones we work with are from the dual of all finite dimensional vectors for mathematical convenience. Of course you cant express the laws of QM without it so my view is a bit contrived.

Thanks
Bill

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Matterwave
Gold Member

Am I imagining something or is the opening line totally 'whaco':
'In this paper we derive a theorem which proves that the physical interpretation implied by the first postulate of quantum mechanics (QM) is inconsistent with the orthodox formalism.'

How can a postulate be inconsistent with the formalism it is part of?

I believe they are saying the "physical interpretation" of the first postulate is what contradicts the orthodox formalism, and not the first postulate itself. I don't know if they mean the first postulate is inconsistent with any physical interpretation or simply the one we ascribe to it via the orthodox formalism (which, iirc, is not very much at all, unless they mean the Copenhagen interpretation as the "orthodox formalism"?).

I have only read the abstract by the way, as I am currently jet-lagged and in no mood to read a philosophy paper on the interpretations of the quantum state.

atyy
The claim of the paper seems rather strong to me, and I don't know whether it is right. But in the context of relativity, I think I can supply an example in the same spirit. The quantum state is defined using a plane of simultaneity. If we only accept invariant quantities as real in relativity, then the quantum state is not real, since simultaneity is not absolute (without specification of a family of observers) in relativity.

Also, when there are sequential measurements, the quantum dynamics includes wave function collapse. The state is Lorentz covariant under changes of inertial frame if time evolution is unitary, but not if collapse is also included. Fortunately, localized events like measurement outcomes and their probabilities are invariant, so those can be considered real, and there is no problem with quantum mechanics and relativity.

Of course, if one removed invariance as a requirement for the physical state, one could have an aether frame, and the quantum state could be FAPP real in that frame.

Matterwave
Gold Member
The claim of the paper seems rather strong to me, and I don't know whether it is right. But in the context of relativity, I think I can supply an example in the same spirit. The quantum state is defined using a plane of simultaneity. If we only accept invariant quantities as real in relativity, then the quantum state is not real, since simultaneity is not absolute (without specification of a family of observers) in relativity.

Also, when there are sequential measurements, the quantum dynamics includes wave function collapse. The state is Lorentz covariant under changes of inertial frame if time evolution is unitary, but not if collapse is also included. Fortunately, localized events like measurement outcomes and their probabilities are invariant, so those can be considered real, and there is no problem with quantum mechanics and relativity.

Of course, if one removed invariance as a requirement for the physical state, one could have an aether frame, and the quantum state could be FAPP real in that frame.

If the Authors' gripe with QM is that it is not Lorentz covariant, then certainly that is not a real gripe? QM is obviously not Lorentz covariant, it was never formulated to be. That's where QFT comes in...

bhobba
atyy
If the Authors' gripe with QM is that it is not Lorentz covariant, then certainly that is not a real gripe? QM is obviously not Lorentz covariant, it was never formulated to be. That's where QFT comes in...

I'm including QFT when I say QM. I think they require real things to be invariant, but the state is defined using simultaneity, and usually we don't consider simultaneity to be invariant.

My remark about collapse is an additional point. In QFT (let's say the Schroedinger functional picture) the state evolution is covariant only for unitary evolution, but not if collapse is included. So the state dynamics is not even covariant (let alone invariant).

vanhees71
Gold Member
2021 Award
First of all: Weyl and von Neumann were mathematicians, not philosophers. Both were partially wrong with the physics part but have helped a lot to understand the mathematical foundations of quantum theory (and in the case of Weyl to General Relativity).

I don't know about the present paper, but when I read the abstract, I'm inclined not to read further, because nobody claims that the quantum state is represented by a Hilbert-space vector. It's very easy to disprove this idea. The most simple argument against this is simply the existence of half-integer spins of particles. It's not Hilbert-space vectors that represent a state but rays in Hilbert space or, equivalently for pure states and including the more general concept of mixed states, statistical operators, i.e., positive semidefinite self-adjoint operators with trace 1.

bhobba
I don't know about the present paper, but when I read the abstract, I'm inclined not to read further, because nobody claims that the quantum state is represented by a Hilbert-space vector. It's very easy to disprove this idea. The most simple argument against this is simply the existence of half-integer spins of particles. It's not Hilbert-space vectors that represent a state but rays in Hilbert space or, equivalently for pure states and including the more general concept of mixed states, statistical operators, i.e., positive semidefinite self-adjoint operators with trace 1.
Maybe you should read section 4 of the paper, Ithink their argument is independent of the formal distinction between rays or statistical operators and vectors in complex Hilbert space.

Demystifier
Gold Member
http://arxiv.org/abs/1412.2701v1

thrashing the idea that a (unit) vector of a Hilbert space can represent the state of a physical quantum system. One of the authors is a philosopher, the other a mathematician and. surprisingly, throughout the paper the mathematics of finite-dim. vector spaces (known to be improper for QM) is used. I don't think, however, that the lack of rigor in maths can be the point which turns their paper from a correct one into a wrong one.

Have a read of it, please, and tell me where they go wrong
I have read the paper and I think I understood it. In my opinion, they are not wrong. This is because their claims are not really so radical as you might think they are. To understand that, one needs to understand carefully what they really mean by verbal expressions such as "physical state".

First, even though they prove their theorems in a finite dimensional space, that's not a problem. Their theorems are akin to the celebrated Kochen-Specker theorem, which is also proved in a finite dimensional space, while nobody considers this to be a problem for the Kochen-Specker theorem.

As I said, the crucial thing is to explain what do they mean by certain verbal expressions. By "physical" they mean "objectively real", and by "objectively real" they mean "basis independent". The last term "basis independent" is mathematically well defined, which allows them to prove rigorous theorems. The identification
physical = objectively real = basis independent
is merely a definition of the concepts, so they are neither right nor wrong about that. In their paper one simply needs to remember that otherwise vague terms "physical" and "objectively real" mean "basis independent", even if in some other papers these vague terms have a different meaning.

Now let me express the content of their theorems in a more common language, in which the words physical and basis independent do not have the same meaning. With such a more common terminology, their theorems say that the physical content of QM is not basis independent. But this claim is not new at all. This is nothing but a restatement of the preferred basis problem appearing in one way or another in all interpretations of QM.

The preferred basis problem is most explicit in the von Neumann collapse interpretation. If the wave function collapses due to observation, in what basis the collapse happens?

Another example is the many-world interpretation. Even though there is no collapse, the preferred basis problem is a very serious one as discussed e.g. in https://www.physicsforums.com/threads/many-worlds-proved-inconsistent.767809/

While it is already known that a preferred basis problem appears in all specific interpretations of QM, in each specific interpretation this problem takes a different form. By contrast, the theorem in the present paper does not assume any specific interpretation, thus presenting a universal interpretation-independent formulation of the preferred basis problem.

This is similar to the status of contextuality in QM. The contextuality takes a different form in different interpretations of QM, while the Kochen-Specker theorem presents a universal interpretation-independent proof of contextuality.

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kith, ShayanJ, TrickyDicky and 4 others
vanhees71
Gold Member
2021 Award
Quantum theory is in fact basis independent. It can be formulated without a basis, as was shown by Dirac in ~1926 and more mathematically rigorously by von Neumann. So I don't understand these statements at all. I also don't think that "collapse" should be part of any interpretation, at least not as a real physical process, because this makes more problems than anything else and is totally unnecessary.

mfb
atyy
Quantum theory is in fact basis independent. It can be formulated without a basis, as was shown by Dirac in ~1926 and more mathematically rigorously by von Neumann. So I don't understand these statements at all. I also don't think that "collapse" should be part of any interpretation, at least not as a real physical process, because this makes more problems than anything else and is totally unnecessary.

It is a matter of definition. If one chooses position to be real, then the real vectors are the eigenvectors of position. This picks out a preferred basis that is distinct from the momentum basis.

vanhees71
Gold Member
2021 Award
But this is not a "preferred basis" but just the choice, which observable I want to measure. If I want to measure position, I'll take the generalized position eigenbasis to evaluate the position-probability distribution; if I want to measure momentum, I take the generalized momentum eigenbasis to evaluate the momentum-probability distribution.

If I then filter according to a position or momentum range, I've prepared a new state. That's a physically meaningful preference but not a "preferred basis" in the sense as if there's a preferred basis in the theory as a whole.

Demystifier
Gold Member
Quantum theory is in fact basis independent. It can be formulated without a basis, as was shown by Dirac in ~1926 and more mathematically rigorously by von Neumann. So I don't understand these statements at all. I also don't think that "collapse" should be part of any interpretation, at least not as a real physical process, because this makes more problems than anything else and is totally unnecessary.
As long as you do not consider measurements, it is true that quantum theory is basis independent. But at that level, it is also physically empty. To give the physical meaning to the quantum theory, you must say something about what happens (or how the formalism has to be used) when a measurement is performed. So can you say something about that? And more importantly, can you say that in a basis independent way? Try to do it and I will tell you how this depends on the basis.

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atyy
But this is not a "preferred basis" but just the choice, which observable I want to measure. If I want to measure position, I'll take the generalized position eigenbasis to evaluate the position-probability distribution; if I want to measure momentum, I take the generalized momentum eigenbasis to evaluate the momentum-probability distribution.

If I then filter according to a position or momentum range, I've prepared a new state. That's a physically meaningful preference but not a "preferred basis" in the sense as if there's a preferred basis in the theory as a whole.

Yes, you choose the preferred basis by your choice of measurement. It is just a matter of definition, not much different from saying that position and momentum cannot be simultaneously real.

Demystifier
Gold Member
But this is not a "preferred basis" but just the choice, which observable I want to measure. If I want to measure position, I'll take the generalized position eigenbasis to evaluate the position-probability distribution; if I want to measure momentum, I take the generalized momentum eigenbasis to evaluate the momentum-probability distribution.

If I then filter according to a position or momentum range, I've prepared a new state. That's a physically meaningful preference but not a "preferred basis" in the sense as if there's a preferred basis in the theory as a whole.
With such an operational view of quantum theory, the point is that it is you who is making the choice which observable to measure. The quantum state itself cannot make such a choice. So you are not the quantum state. But you are certainly physical, so there is something physical which is not a quantum state. Furthermore, as you are not a quantum state, you do not live in the Hilbert space and in that sense you cannot say that you are "basis independent". So there is something physical which is not basis independent.

You might say that you are not basis dependent either, simply because you do not live in the Hilbert space implying that basis dependence/independence is simply not a property of you. That's correct, but there is still something basis dependent about you, because you choose one basis or another by choosing one observable or another to measure. A choice of an obserbable corresponds to a choice of a basis, because each observable defines a preferred basis - the one in which this observable is diagonal.

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ShayanJ and vanhees71
atyy
As long as you do not consider measurements, it is true that quantum theory is basis independent. But at that level, it is also physically empty. To give the physical meaning to the quantum theory, you must say something about what happens (or how the formalism has to be used) when a measurement is performed. So can you say something about that? And more importantly, can you say that in a basis independent way? Try to do it and will tell you how this depends on the basis.

Although their narrow conclusion is (probably) strictly correct, I wonder whether their wider discussion, such as their criticism of PBR etc is good or not. PBR uses a hidden variable framework, which includes Bohmian Mechanics. Their definition of real doesn't seem to impact at all whether the wave function is "real" in Bohmian Mechanics (where it can be considered real), and the basis dependence is (I think) just contextuality.

Demystifier
Gold Member
Although their narrow conclusion is (probably) strictly correct, I wonder whether their wider discussion, such as their criticism of PBR etc is good or not. PBR uses a hidden variable framework, which includes Bohmian Mechanics. Their definition of real doesn't seem to impact at all whether the wave function is "real" in Bohmian Mechanics (where it can be considered real), and the basis dependence is (I think) just contextuality.
Well, the point is that they use a different definition of "reality" than PBR (and Bohmians) do. As I said, they define "real" as "basis independent", which is not the definition of reality according to PBR or Bohmians. Thus, they are right that the state is "non-real" according to their definition, but PBR are also right that the state is "real" according to the PBR definition.

vanhees71
Gold Member
2021 Award
According to quantum theory I am a quantum state, which behaves pretty classically, because I'm in exchange with the environment and consist of a large number of micro- states making up my macrostate, but that's another topic.

The state of the quantum system is of course also choosen by me as an observer. If I'd be an experimentalist I'd even actively use some equipment to prepare quanta in a specific state, like the particle physicists at the LHC do when they prepare to proton beams banging head on at nearly 14 GeV center-mass energy pretty soon (hopefully). I don't see, where there should be a problem with a preferred basis. It's simply the state determination as described carefully, e.g., in Ballentine's book, i.e., within standard quantum theory in the minimal interpretation (which I also dubbed the "no-nonsense interperation" once :-)).

atyy
Well, the point is that they use a different definition of "reality" than PBR (and Bohmians) do. As I said, they define "real" as "basis independent", which is not the definition of reality according to PBR or Bohmians. Thus, they are right that the state is "non-real" according to their definition, but PBR are also right that the state is "real" according to the PBR definition.

Yes, I understood that their claim is probably right after reading your analysis in post #10. The interesting thing is they also say things like (p17) "As we have proven through the NDI theorem, the PBR theorem is derived from a false hypothesis making its result untenable." Presumably that is not a correct criticism of PBR then?

Demystifier
Gold Member
Yes, I understood that their claim is probably right after reading you analysis in post #10. The interesting thing is they also say things like (p17) "As we have proven through the NDI theorem, the PBR theorem is derived from a false hypothesis making its result untenable." Presumably that is not a correct criticism of PBR then?
I agree with you, it's not a correct criticism of PBR. What they seem to miss is that their identification
real = basis independent
is not a true fact, but merely a convenient definition.

I agree with you, it's not a correct criticism of PBR. What they seem to miss is that their identification
real = basis independent
is not a true fact, but merely a convenient definition.
But what they are denouncing is the contradiction between the mathematical formalism(basis dependence of the states) and
the first postulate of QM. That kind of double talk warrants confusion and halts progress.

vanhees71
Gold Member
2021 Award
There's no basis dependence in the mathematical formalism to begin with. The postulate about the states is wrong (see my first posting in this thread). So why the heck should we analyze the paper further?

Demystifier
Gold Member
According to quantum theory I am a quantum state, which behaves pretty classically, because I'm in exchange with the environment and consist of a large number of micro- states making up my macrostate, but that's another topic.

The state of the quantum system is of course also choosen by me as an observer. If I'd be an experimentalist I'd even actively use some equipment to prepare quanta in a specific state, like the particle physicists at the LHC do when they prepare to proton beams banging head on at nearly 14 GeV center-mass energy pretty soon (hopefully). I don't see, where there should be a problem with a preferred basis. It's simply the state determination as described carefully, e.g., in Ballentine's book, i.e., within standard quantum theory in the minimal interpretation (which I also dubbed the "no-nonsense interperation" once .
First, by saying that you exchange something with the environment contains a preferred basis problem. Namely, the split of the whole system into "you" and "enviroment" depends on the choice of basis for the whole system.

Second, you say that
i) you are a quantum state, and
ii) you have an ability to choose another state
Is that consistent? If so, then a state has ability to choose another state. But how that ability is realized? Can it be described by the Schrodinger equation alone? If yes, then you are probably assuming a many-world interpretation, for which it is well-known to lead to the preferred basis problem. If not, then you probably need some other equation, but then what that other equation is, and are you sure that it is basis independent? These are all non-trivial questions, and whatever your answer is, I claim that the preferred basis problem emerges. If you do not see it, try to answer all these questions; depending on your answer I will tell you how the preferred basis problem then emerges.

Third, note that Ballentine would not agree that you are the quantum state.

Demystifier
Gold Member
But what they are denouncing is the contradiction between the mathematical formalism(basis dependence of the states) and
the first postulate of QM. That kind of double talk warrants confusion and halts progress.
The states do not depend on the basis and they do not say they do. What they show is that physics depends on the basis, implying that the state by itself is not physics.

atyy
If so, then a state has ability to choose another state. But how that ability is realized? Can it be described by the Schrodinger equation alone? If yes, then you are probably assuming a many-world interpretation, for which it is well-known to lead to the preferred basis problem.

Third, note that Ballentine would not agree that you are the quantum state.

Many interpreters of Ballentine are secretly MWI, as noticed by Laloe also :)

"In fact, experience shows that defenders of the correlation point of view, when pressed hard in a discussion to describe their point of view with more accuracy, often express themselves in terms that come very close to the Everett interpretation (see § 6.5); in fact, they may sometimes be proponents of this interpretation without realizing it!" http://arxiv.org/abs/quant-ph/0209123 (p68)

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Demystifier
Gold Member
The postulate about the states is wrong (see my first posting in this thread). So why the heck should we analyze the paper further?
Because all their theorems can easily be restated by replacing "states" with rays in the Hilbert space.

stevendaryl
Staff Emeritus
There's no basis dependence in the mathematical formalism to begin with. The postulate about the states is wrong (see my first posting in this thread). So why the heck should we analyze the paper further?

I'm a little confused by the claims of basis dependence, but I'm not positive that there isn't any.

At an abstract enough level, quantum mechanics is independent of basis. But to make the connection with observations, you have to perform measurements. And to do that, you have to say which observable your macroscopic measuring device is measuring.

That's the part that seems to not be specified by the QM formalism. Roughly speaking, you need "pointer states": states of the macroscopic device that are
1. correlated with the observable you're interested in.
2. different enough, macroscopically, that you can check which state the device is in.
I don't know a way to say that a device measures a particular observable without invoking the macroscopic/microscopic distinction, which isn't part of the mathematics of quantum mechanics.

atyy
I'm a little confused by the claims of basis dependence, but I'm not positive that there isn't any.

It's a little bit like simultaneity in special relativity. Is simultaneity absolute? Yes, relative to a family of observers. Similarly, the eigenvectors of an observable are absolute, relative to an observable.

stevendaryl
Staff Emeritus
It's a little bit like simultaneity in special relativity. Is simultaneity absolute? Yes, relative to a family of observers. Similarly, the eigenvectors of an observable are absolute, relative to an observable.

My question was what it means to say that a macroscopic device measures observable $X$. As I said, that seems to necessarily involve making a macroscopic/microscopic distinction that goes beyond the mathematics of quantum mechanics.

atyy
My question was what it means to say that a macroscopic device measures observable $X$. As I said, that seems to necessarily involve making a macroscopic/microscopic distinction that goes beyond the mathematics of quantum mechanics.

Can't we understand macroscopic and microscopic to be fundamental undefined concepts that are part of the mathematics? So macroscopic is a synonym for observable, and microscopic is a synonym for quantum state.

kith
I don't know a way to say that a device measures a particular observable without invoking the macroscopic/microscopic distinction, which isn't part of the mathematics of quantum mechanics.
I think this is the only part of the problem which is widely considered to be solved.

As far as I can see, there are three problems associated with the preferred basis:
1) the factorization problem
2) the actual problem of the preferred basis
3) the problem of outcomes

The factorization problem notes that in order to picture the measurement as a quantum interaction between the system and the apparatus leading to entanglement between the two, you need to decompose the big Hilbert space in a certain way. If you have only the big Hilbert space and the full Hamiltonian, there are always bases where you only have simple phase rotations. So you don't get the picture of interacting subsystems by these two entities alone. I think this is what Demystifier is talking about when he says "MWI has a well-known preferred basis problem". There's a paper on all this by Schwindt.

The actual problem of the preferred basis is the question if and how the eigenstates of an observable are singled out dynamically in a measurement. This is what you are referring to and I think this is what Zurek's environmental induced superselection or decoherence explains by using only quantum dynamics (and certain approximations) for the composite system. A device measures a particular observable because its Hamiltonian and its thermodynamical properties lead to a maximal and robust entanglement between the eigenstates of the observable for the system and the states of the pointer of the device.

The problem of outcomes is the question why a single outcome is observed although the final state after decoherence is a mixed state.

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Demystifier
The states do not depend on the basis and they do not say they do. What they show is that physics depends on the basis, implying that the state by itself is not physics.
Right, the state mentioned in the first QM postulate is supposed according to its wording to be basis-independent and to be physical. They show that is contradicted by the math, and therefore the physics, that is in practice basis-dependent in the way you explain( but they use a different path to prove their theorem).

Demystifier